On the equivalence of Legendrian and transverse invariants in knot Floer homology

TL;DR

This paper proves the conjectured equivalence of different invariants of transverse knots in contact 3-manifolds by introducing a new invariant that aligns with existing ones, unifying their understanding.

Contribution

The authors define a new transverse knot invariant in knot Floer homology and demonstrate its equivalence with previously known invariants, confirming a key conjecture.

Findings

The new invariant agrees with Ozsvath, Szabo, and Thurston's invariant in the tight contact 3-sphere.

The new invariant matches the Lisca, Ozsvath, Stipsicz, and Szabo invariant in general contact 3-manifolds.

The equivalence confirms the conjecture about the invariants' consistency across different settings.

Abstract

Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.